Subgroups — C₂⁴×C₅×AGL(1,𝔽₇) #72218251

Order 3360.

2992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×AGL(1,𝔽₇)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₅×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×C₅×ΓL(1,𝔽₈)	C₂²	1	15	yes	complemented
5	C₂²×C₅×AGL(1,𝔽₇)	C₂²	1	140	yes	direct product factor
6	C₂⁴×AGL(1,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁴×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂³×C₅×D₇	C₂×C₃	1	30	yes	complemented
9	C₂⁵×C₃×C₅	—	7	1	no	maximal, Hall
10	C₂²×C₅×ΓL(1,𝔽₈)	C₂³	1	35	yes	complemented
11	C₂×C₅×AGL(1,𝔽₇)	C₂³	1	120	yes	direct product factor
12	C₂⁴×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented
13	C₂³×AGL(1,𝔽₇)	C₂×C₅	1	30	yes	direct product factor
14	C₂³×C₅×C₇	C₂²×C₃	1	15	yes	complemented
15	C₂²×C₅×D₇	C₂²×C₃	1	140	yes	complemented
16	C₂⁴×C₃×C₅	—	7	30	no	—
17	C₂⁴×C₃×C₅	—	7	1	no	—
18	C₂⁴×D₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
19	C₂×C₅×ΓL(1,𝔽₈)	C₂⁴	1	15	yes	complemented
20	C₅×AGL(1,𝔽₇)	C₂⁴	1	16	yes	direct product factor
21	C₂³×ΓL(1,𝔽₈)	C₂²×C₅	1	15	yes	complemented
22	C₂²×AGL(1,𝔽₇)	C₂²×C₅	1	140	yes	direct product factor
23	C₂⁵×C₅	—	7	1	no	Hall
24	C₂²×C₅×C₇	C₂³×C₃	1	35	yes	complemented
25	C₂×C₅×D₇	C₂³×C₃	1	120	yes	complemented
26	C₂³×C₃×C₅	—	7	140	no	—
27	C₂³×C₃×C₅	—	7	15	no	—
28	C₂⁴×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
29	C₂³×D₇	C₂×C₃×C₅	1	30	yes	complemented
30	C₅×ΓL(1,𝔽₈)	C₂⁵	1	1	yes	characteristic, complemented, Hall
31	C₂⁵×C₃	—	7	1	no	Hall
32	C₂²×ΓL(1,𝔽₈)	C₂³×C₅	1	35	yes	complemented
33	C₂×AGL(1,𝔽₇)	C₂³×C₅	1	120	yes	direct product factor
34	C₂⁴×C₅	—	7	30	no	—
35	C₂⁴×C₅	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
36	C₂×C₅×C₇	C₂⁴×C₃	1	15	yes	complemented
37	C₅×D₇	C₂⁴×C₃	1	16	yes	complemented
38	C₂²×C₃×C₅	—	7	120	no	—
39	C₂²×C₃×C₅	—	7	35	no	—
40	C₂³×C₇	C₂²×C₃×C₅	1	15	yes	complemented
41	C₂²×D₇	C₂²×C₃×C₅	1	140	yes	complemented
42	C₂⁴×C₃	—	7	30	no	—
43	C₂⁴×C₃	—	7	1	no	—
44	C₂×ΓL(1,𝔽₈)	C₂⁴×C₅	1	15	yes	complemented
45	AGL(1,𝔽₇)	C₂⁴×C₅	1	16	yes	direct product factor
46	C₂³×C₅	—	7	140	no	—
47	C₂³×C₅	C₂×AGL(1,𝔽₇)	1	15	yes	direct product factor
48	C₅×C₇	C₂⁵×C₃	1	1	yes	characteristic, complemented, Hall
49	C₂⁵	—	7	1	no	Sylow
50	C₂×C₃×C₅	—	7	16	no	—
51	C₂×C₃×C₅	—	7	15	no	—
52	C₂²×C₇	C₂³×C₃×C₅	1	35	yes	complemented
53	C₂×D₇	C₂³×C₃×C₅	1	120	yes	complemented
54	C₂³×C₃	—	7	140	no	—
55	C₂³×C₃	—	7	15	no	—
56	ΓL(1,𝔽₈)	C₂⁵×C₅	1	1	yes	characteristic, complemented, Hall
57	C₂²×C₅	—	7	120	no	—
58	C₂²×C₅	C₂²×AGL(1,𝔽₇)	1	35	yes	direct product factor
59	C₂⁴	—	7	30	no	—
60	C₂⁴	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
61	C₃×C₅	—	7	1	no	Hall
62	C₂×C₇	C₂⁴×C₃×C₅	1	15	yes	complemented
63	D₇	C₂⁴×C₃×C₅	1	16	yes	complemented
64	C₂²×C₃	—	7	120	no	—
65	C₂²×C₃	—	7	35	no	—
66	C₂×C₅	—	7	16	no	—
67	C₂×C₅	C₂³×AGL(1,𝔽₇)	1	15	yes	direct product factor
68	C₂³	—	7	140	no	—
69	C₂³	C₂×C₅×AGL(1,𝔽₇)	1	15	yes	direct product factor
70	C₇	C₂⁵×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	16	no	—
72	C₂×C₃	—	7	15	no	—
73	C₅	C₂⁴×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	7	120	no	—
75	C₂²	C₂²×C₅×AGL(1,𝔽₇)	1	35	yes	direct product factor
76	C₃	—	7	1	no	Sylow
77	C₂	—	7	16	no	—
78	C₂	C₂³×C₅×AGL(1,𝔽₇)	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴×C₅	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂⁵×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴×C₅

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×C₅×ΓL(1,𝔽₈)	C₂	semidirect
2. C₂³×C₅×AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂⁴×C₅×D₇	C₃	semidirect
4. C₂³×C₅×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂²×C₅×AGL(1,𝔽₇)	C₂²	direct (N × H)
6. C₂⁴×AGL(1,𝔽₇)	C₅	direct (N × H)
7. C₂⁴×C₅×C₇	C₂×C₃	semidirect
8. C₂³×C₅×D₇	C₂×C₃	semidirect
10. C₂²×C₅×ΓL(1,𝔽₈)	C₂³	semidirect
11. C₂×C₅×AGL(1,𝔽₇)	C₂³	direct (N × H)
12. C₂⁴×ΓL(1,𝔽₈)	C₂×C₅	semidirect
13. C₂³×AGL(1,𝔽₇)	C₂×C₅	direct (N × H)
14. C₂³×C₅×C₇	C₂²×C₃	semidirect
15. C₂²×C₅×D₇	C₂²×C₃	semidirect
18. C₂⁴×D₇	C₃×C₅	semidirect
19. C₂×C₅×ΓL(1,𝔽₈)	C₂⁴	semidirect
20. C₅×AGL(1,𝔽₇)	C₂⁴	direct (N × H)
21. C₂³×ΓL(1,𝔽₈)	C₂²×C₅	semidirect
22. C₂²×AGL(1,𝔽₇)	C₂²×C₅	direct (N × H)
24. C₂²×C₅×C₇	C₂³×C₃	semidirect
25. C₂×C₅×D₇	C₂³×C₃	semidirect
28. C₂⁴×C₇	C₂×C₃×C₅	semidirect
29. C₂³×D₇	C₂×C₃×C₅	semidirect
30. C₅×ΓL(1,𝔽₈)	C₂⁵	semidirect
32. C₂²×ΓL(1,𝔽₈)	C₂³×C₅	semidirect
33. C₂×AGL(1,𝔽₇)	C₂³×C₅	direct (N × H)
35. C₂⁴×C₅	AGL(1,𝔽₇)	direct (N × H)
36. C₂×C₅×C₇	C₂⁴×C₃	semidirect
37. C₅×D₇	C₂⁴×C₃	semidirect
40. C₂³×C₇	C₂²×C₃×C₅	semidirect
41. C₂²×D₇	C₂²×C₃×C₅	semidirect
44. C₂×ΓL(1,𝔽₈)	C₂⁴×C₅	semidirect
45. AGL(1,𝔽₇)	C₂⁴×C₅	direct (N × H)
47. C₂³×C₅	C₂×AGL(1,𝔽₇)	direct (N × H)
48. C₅×C₇	C₂⁵×C₃	semidirect
52. C₂²×C₇	C₂³×C₃×C₅	semidirect
53. C₂×D₇	C₂³×C₃×C₅	semidirect
56. ΓL(1,𝔽₈)	C₂⁵×C₅	semidirect
58. C₂²×C₅	C₂²×AGL(1,𝔽₇)	direct (N × H)
60. C₂⁴	C₅×AGL(1,𝔽₇)	direct (N × H)
62. C₂×C₇	C₂⁴×C₃×C₅	semidirect
63. D₇	C₂⁴×C₃×C₅	semidirect
67. C₂×C₅	C₂³×AGL(1,𝔽₇)	direct (N × H)
69. C₂³	C₂×C₅×AGL(1,𝔽₇)	direct (N × H)
70. C₇	C₂⁵×C₃×C₅	semidirect
73. C₅	C₂⁴×AGL(1,𝔽₇)	direct (N × H)
75. C₂²	C₂²×C₅×AGL(1,𝔽₇)	direct (N × H)
78. C₂	C₂³×C₅×AGL(1,𝔽₇)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₅×ΓL(1,𝔽₈)	2
C₂³×C₅×AGL(1,𝔽₇)	2
C₂⁴×C₅×D₇	3
C₂⁴×AGL(1,𝔽₇)	5
C₂⁵×C₃×C₅	7

Subgroups — C24×C5×AGL(1,𝔽7) #72218251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₅×AGL(1,𝔽₇) #72218251